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\chapter*{Abbreviations and Symbols}

\section*{Abbreviations}

\begin{tabular}{ll}
%BVLS & \\
%CSVM & Central Support Vector Machine \\
EDR & Error-Dependent Repetition \\
ERM & Empirical Risk Minimization \\
iid & Independent and Identically Distributed \\
KKT & Karush-Kuhn-Tucker \\
KM  & $k$-means \\
LP  & Linear Programming \\
MIPS & Million of Instructions per Second \\
MFLOPS & Million of Floating Point Instructions per Second \\
NC  & Number of Clusters \\
%NPA & Nearest Point Algorithm \\
PCA & Principal Component Analysis \\
PDF & Probability Density Function \\
QP  & Quadratic Programming \\
RBF & Radial Basis Function \\
SMO & Sequential Minimal Optimization \\
SOR & Successive Over Relation \\
SRM & Structural Risk Minimization \\
SV  & Support Vector \\
LM  & Lagrange multiplier \\
SVM & Support Vector Machine \\
SVD & Singular Value Decomposition \\
VC  & Vapnik-Chervonenskis \\
$O(\cdot)$ & Order of \\
$\log$ & Logarithm \\
$\mathrm{max}$ & Maximum \\
$\mathrm{min}$ & Minimum \\
$\mathrm{mod}$ & Module \\
\end{tabular}

\section*{Symbols}

\begin{tabular}{ll}
                   & {\bf General conventions }\\
                   & \\
$y$ 		           & An ordinary scalar  \\
$\mathbf{a}$ 	     & Vector $\mathbf{a}$ with dimension $n \times 1$ \\
$\mathbf{A}$ 	     & Matrix $\mathbf{A}$ with dimension $n \times m$ \\
$\mathbf{1}$ 	     & A vector of ones with dimension $n \times 1$ \\
$\mathbf{I}$ 	     & Identity matrix with dimension $n \times n$ \\
$\mathbf{x}^T$     & Transpose of vector $\mathbf{x}$ \\
$|x|$              & Absolute value of $x$ \\
$\|\mathbf{x}\|$   & Euclidean norm of $\mathbf{x}$ \\
$p(x)$             & Probability of $x$  \\
$p(x|y)$           & Conditional probability of $x$ given $y$ \\
$L(\cdot)$         & Loss function \\
$R(\cdot)$         & Risk functional \\
                   & \\
                   & {\bf Specific conventions }\\
                   & \\
                   %%% ---  SVM ----
$\mathrm{x}_i$     & $i$-th input pattern \\
$p$                & Number of input patterns \\
$m$                & Input pattern dimension \\
$y_i$ 		         & Output for pattern $\mathbf{x}_i$  \\
$\alpha_i$         & $i$-th Lagrange multiplier\\
$f(\mathbf{x}_i)$  & SVM output for pattern $\mathbf{x}_i$  \\
$w_i$ 		         & $i$-th weight  \\
$\varphi_i(\cdot)$ & $i$-th mapping function \\
$b$                & Bias \\
$\xi_i$            & $i$-th slack variable \\
$C$                & Upper limit for Lagrange multipliers \\
$K(\cdot,\cdot)$   & Kernel function \\
							     %%% ---  KM ----
$\omega$           & State of nature \\
$k$                & Number of cluster for $k$-means algorithm \\
$\lambda_i$        & ``Mixness'' degree for pattern $\mathbf{x}_i$ \\
$S_i$              & $i$-th disjoint subset (cluster) \\
$S_i^{(t)}$        & $i$-th disjoint subset (cluster) at iteration $t$ \\
\end{tabular}

\begin{tabular}{ll}
                   & {\bf Specific conventions } (continued)\\
                   & \\
$\mathbf{c}_i$     & Center of cluster $S_i$  \\
$\mathbf{c}_i^{(t)}$ & Center of cluster $S_i^(t)$  at iteration $t$ \\
$f_d(\mathbf{x},\mathbf{y})$ & Distance between vectors $\mathbf{x}$ and $\mathbf{y}$ \\
$d$                & Input pattern dimension after dimensionality reduction \\
$\varepsilon$      & Small positive constant \\
$t_h$              & Threshold value for feature selection \\                   
							     %%% --- EDR ----
$\mathcal X$       & Input space \\
$\mathcal Y$       & Output space \\
$h_t$              & Hypothesis at iteration $t$, a prediction rule on $\mathcal X$ \\
$1-\epsilon$       & Hypothesis accuracy  ($\epsilon$ is, ideally, an small constant) \\
$1-\delta$         & Hypothesis reliability  ($\delta$ is, ideally, an small constant) \\
$D_t(i)$           & Probability distribution for pattern $x_i$ at iteration $t$ \\
$\epsilon_t$       & Hypothesis error at iteration $t$\\
$a_t$              & Weight for hypothesis $h_t$ at iteration $t$\\
$e_i$ 		         & Output error for pattern $\mathbf{x}_i$  \\
$n_E$              & Number of EDR's scans \\
$\mathcal{E}_t$    & Ordered set of errors, at iteration $t$ \\
$\mathcal{E}_t^E$  & Ordered set of errors, at iteration $t$, after applying \\
                   & comparison function \\
$m_t^E$            & Size of set $\mathcal{E}_t^E$ set at iteration $t$ \\
$d_t(i)$           & Number of presentations at iteration $t$ for pattern $i$ \\
$p_E$              & Comparison function power \\
$g_t(e)$           & Error distribution probability at iteration $t$ \\
\end{tabular}